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In this video, the speaker dives into the concept of infinity and how it can be understood through the open and closed square example and the concepts of infinitely large and small. They explore hyper reals, logistic functions, transfer principle, and geometric algebra—allowing for infinitely small and large numbers and the ability to take derivatives and limits not possible before. They reference a book on probability theory which uses only the concepts of infinitely large and small and covers more than the speaker has ever done in their lifetime.

Infinitely large numbers, represented by n and epsilon, can be used to derive the Taylor series for e to the x. Using the binomial theorem, terms can be cancelled and the nested product of n times n minus 1 times n minus 2 and so on can be simplified. Infinity minus one divided by Infinity is equal to a classic formula, where K is a standard limited number. Additionally, the real line has two additional bits attached to it, one being a tiny copy of a line in between all standard points, and the other being infinitesimal neighborhoods. A quiz is given to determine which expressions are standard, such as one minus 0.999 repeating and epsilon, which is smaller than any standard number. Unlimited numbers are bigger than any standard number.

Omega and Omega over two are infinite numbers that behave like finite numbers and can be used to calculate precise answers. Omega squared has a square root of Omega, and these hyper real numbers can be used to prove that there are unlimited prime numbers. William Blake's poem is used to illustrate how even small things can contain entire worlds. Infinitely close approximations of points can be used to zoom in to a line and create a new line with different properties. This process can be repeated infinitely, creating a new line with each iteration that shares finitary properties with normal numbers, but has second order properties that are not captured by first order logic. This relationship between infinitely close points allows for equality to be proven in a single step.

Standard part function is a method of rounding off non-standard limited numbers to a standard real number. The step function is zero for negative inputs, one for positive inputs and one half at the point zero. To model this, a logistic curve is used which is infinitely close to the step function at all the standard points. The exponential expression e to the minus nx is strictly positive and not infinitesimal. This allows for the idea of a number being close to another number, even though there is no such thing in the standard reals. Hyper reals extend the class of differentiability, meaning that any non-differentiable functions on the reals can be extended to the hyper reals and a continuously differentiable function can be found that is infinitely close to the original function.

Logistic functions can be differentiated to create a hyper-finite mesh which captures everything. This derivative is similar to a normal distribution, with an infinitesimal variance and an infinite height at zero. Differentiating again results in a dipole moment resembling a flower. Euclid's Elements introduced the concept of a horn angle, which is an angle smaller than any other angle. Probabilities of hitting a number exactly in a closed unit interval are zero, and can be represented by Epsilon. Hyperreals allow for infinitely small and large numbers and the ability to take derivatives and limits in a way not possible before. This enlargement of numbers is argued to be a natural step and provides stronger intuitions about numbers.

Transfer principle is a way of discretizing the continuum into infinite sequences of real numbers. Hyperfinite approximation is an example of this, where a smooth curve is made into an infinitesimal chain with an unlimited number of links. Infinite numbers can be related to other quantities that are also infinite, such as circles, which have an unlimited number of points. Euclid's axioms can be used to construct a polygon, and computer science has explored hyper finite proofs, which involve taking a finite construction and taking it to its limit. These are useful for those working in differential equations and ultrafinite tests.

The speaker talks about the concept of a number as a way to unify discreet and continuous systems, and how infinity can be understood through the open and closed square example. They reference a book on probability theory which uses only the concepts of infinitely large and small and covers more than the speaker has ever done in their lifetime. They discuss how computers can use Ultra filters and syntactic approaches to model hyper reels, and how Geometric algebra can be used to model cyclic orders. They are skeptical of the book's claims, but acknowledge that it is a rigorous treatment and covers a lot of material, such as stochastic processes.

Infinitely large numbers, represented by n and epsilon, can be used to derive the Taylor series for e to the x. By dropping the limit and allowing n to be an infinitely large positive integer, a massive polynomial is created. Using the binomial theorem, terms can be cancelled and the nested product of n times n minus 1 times n minus 2 and so on can be simplified. This allows the Taylor series to be derived using the fact that n is infinite.

Infinity minus one divided by Infinity is equal to a classic formula, where K is a standard limited number. Expanding out the terms shows that the expression is infinitesimal, meaning it is infinitely close to one but not quite one. The real line has two additional bits attached to it, one being a tiny copy of a line in between all standard points, and the other being infinitesimal neighborhoods which are sets of points that are infinitesimally far from a given one.

A quiz is given involving Epsilon, 0.999 repeating, E, and Pi to determine what is considered standard. Epsilon minus 2 is not standard because it involves a funny number that is not a limited real number. Tangent of Big N is also not standard, as Big N is not a standard expression. One minus 0.999 repeating is considered standard, as it is close to, but not quite one. Epsilon is smaller than any standard number, and unlimited numbers are bigger.

Omega and Omega over two are examples of infinite numbers that act like they are finite. They behave like ordinary numbers and can be used to calculate precise answers, such as Omega minus Omega over three being equal to two-thirds Omega. Omega squared is just Omega squared and the square root of Omega squared is Omega. These hyper real numbers can be used to prove that there are unlimited prime numbers, not just infinitely many. William Blake's poem is used to demonstrate that even the smallest things can contain entire worlds.

Infinitely close approximations of points can be used to zoom in to a line and create a new line with different properties. This process can be repeated infinitely, creating a new line with each iteration. These new lines share finitary properties with normal numbers, but possess second order properties that are not captured by first order logic. This relationship between infinitely close points allows for equality to be proven in a single step, rather than proving inequalities twice.

The speaker discusses the concept of standard part function, which involves rounding off non-standard limited numbers to a standard real number by dropping epsilons. They then discuss the step function, which is zero for negative inputs, one for positive inputs, and one half at the point zero. To model this, they use a logistic curve, which is infinitely close to the step function at all the standard points. The table is a case analysis, and the multiplication by n highlighted in red is where the magic happens.

The exponential expression e to the minus nx is strictly positive and not infinitesimal. For positive inputs, plugging in a standard number such as 2 results in a value close to one, while negative inputs result in a value close to zero. Zero is the only number which is both standard and infinitesimal, and any value between zero and one when x is infinitesimally close to zero simplifies to one over one plus one over n.

Infinitely close values can be used to create a grid of points, which allows for the idea of a number being close to another number, even though there is no such thing in the standard reals. The hyper reals extend the class of differentiability, meaning that any non-differentiable functions on the reals can be extended to the hyper reals, and a continuously differentiable function can be found that is infinitely close to the original function. This is possible due to the statement that the rationals are dense in the reals, and the hyper rationals are infinitely close to every real number.

Logistic functions can be differentiated to create a hyper-finite mesh which captures everything. The derivative of the function is similar to a normal distribution, with an infinitesimal variance and an infinite height at zero. Differentiating again results in a dipole moment, which looks like a flower. Physicists care a lot about this direct distribution and its derivatives, but there is no general way to multiply them.

In Euclid's Elements, a horn angle is an angle smaller than any other angle, but still present. The probability of hitting a number exactly in a closed unit interval when throwing a dart is zero, as the points have tiny lines around them. This can be divided into infinitesimal intervals, with the probability being represented by Epsilon. This is due to all probabilities rounding to zero in the standard system.

Hyperreals are a way of enlarging the idea of numbers, allowing for infinitely small and large numbers and the ability to distinguish scenarios. This enlargement of numbers allows for derivatives and limits to be taken in a way that was not possible before. It is argued that this is a natural step and that it allows for stronger intuitions about numbers. There is some arbitrariness in the choice of infinite or infinitesimal number but this is also the case in the standard theory.

Transfer principle is a useful way of thinking about hyperreal numbers, where they are compiled into infinite sequences of real numbers. Hyperfinite approximation is an example of how this discretizes the continuum, where given a smooth curve, it can be made into an infinitesimal chain with an unlimited number of links. Questions that are bigger than any standard number cannot be answered, and this outer realm is where research in the overall field happens. People who use this in practice are those who work in differential equations and ultrafinite tests.

Infinite numbers can be related to other quantities that are also infinite. For example, a circle can be made up of an unlimited number of points. A circle with twice the radius of the original would have twice as many points. Euclid's axioms can be used to construct a polygon, which can be thought of as an algorithm. Computer science has explored the concept of hyper finite proofs, which involve taking a finite construction and taking it to its limit.

Sylvia van mackers is one example of an infinite ethical number. Computers can use Ultra filters and syntactic approaches to model hyper reels. Constructive hyper rails use a weak form of choice and have a theory of constructive hyper rails. There is an existing theory of hyper finite proof which uses increasing sequences of finite numbers to prove that a problem is undecidable. Dan Roy and House have used this in statistics. Geometric algebra is a system of beliefs which can be used to model cyclic orders.

The speaker talks about the unification of discreet and continuous systems through the concept of a number, and how it can be used to compress information. They mention how infinity can be understood through the open and closed square example, and how it relates to probability theory. They reference a book on probability theory which uses only the concepts of infinitely large and small and covers more than the speaker has ever done in their lifetime. They are skeptical of the book's claims, but acknowledge that it is a rigorous treatment and covers a lot of material, such as stochastic processes.

Linear logic is an interesting field of mathematics that explores the concept of infinite potential. It looks at how an unlimited supply of something can be realized through interaction, as opposed to a finite amount. The tricky part is understanding the mathematical content of linear logic. It attempts to capture the idea that while the ultimate number of forms of something is limited, it can still be much larger than any number previously named. This is achieved through the compactness theorem, which quantifies over the elements being discussed and picks something bigger than all of them and all possible combinations of them.

hello today I'm going to explore one Topic in particular and yeah the main selling point is that you'll have a consistent way for talking about the idea of very big and for this entire talk big n and Epsilon which is 1 over n are going to represent the same number for every slide specifically some infinitely large integer and I'll explain what infinitely large means but for now really just think of it like any other number that's just it's bigger than one it's bigger than two it's bigger than three and all the normal numbers and then Epsilon is smaller than one it's smaller than a half a third AKA infinitesimal so what does this concept of an infinitely big number bias well here's one thing a somewhat simplified derivation of the tailored series for e to the x starting with the definition from continuous compound interest where the rate in the compounding would be the X at the very top left and then n is the number of compounding periods the usual way this is set up is you take a limit as n the number of periods grows and grows and you'd end up with e to the x but we're going to do it in a different way so the color coding the red means that first we just get rid of this limit we just drop it instead we allow this n to just be the N I had talked about an infinitely large positive integer okay now that I've done this this is some massive polynomial with an infinitely large number of terms but it is a polynomial and using the binomial theorem I can rewrite it this bit in red of 1 to the power of n minus k EPT it is just one to the power of some integer n is therefore just one and multiplying by it doesn't affect anything so we can drop the whole thing leaving behind n choose K and X over n all to the power of K which I just expand out in this line the next step is to look for more stuff to cancel because already you can kind of see the Taylor series you might want to move to the side a little your your head is in the way of the calculation like in real life or in your virtual head there yep okay perfect okay so I cancel out just using definition of factorial part of this numerator and the whole of this denominator of n minus K factorial which leaves behind n times n minus 1 times n minus 2 and so on from n minus K plus 1. so this whole top thing has exactly K terms and now for where we use the fact that n is infinite in an essential way I can rewrite this green bit divided by n over K as the red bit just below it the nested product and observe the note to the side the

idea is that since n is infinitely big think of the intuitive expression in Infinity minus one all divided by Infinity with the infinity is supposed to represent the same quantity it's infinitely close to one it is not quite one but it is infinitely close to it and so we can just cancel the whole thing for y That's a legit thing to do in more depth we'll get into it but now this has gone leaving behind just this which you can see is the classic formula and if Hiro is using the fact that K is just a standard limited number for K that counts up to some unlimited index the expanding out the terms will give you that you're dividing some unlimited number by some unlimited number squared and therefore that this whole expression is infinitesimal that's what the squiggly zero means and can be dropped and this is why I can just say let this specific infinite number be the upper index that I'm counting like why didn't I just use 2N or 4N or n to the power of 40. well I could have then it would make a difference for unlimited K but the difference is infinitesimal so that's why I write the squiggly in the blue boxed area for e to the x is the standard part of this unlimited sum and if you write it to just include standard numbers then it's equal to it okay questions was all of this familiar not with this funny sort of infinite end no but the limit form of that is okay so funny infinite end fair enough should I move throughout this talk I start on the more intuitive end and then build towards more formal definitions the aim of this talk is more to give an idea for how to actually use this concept of an infinite number because there are many resources online to look up the formal construction of them and I'd be happy to talk about all of them in the later half all right here is a picture of the implicit structure we're working with the top line here that I boxed in blue just now you guys do see the unboxing it right so the real line that we're all familiar with is still there but now it has these two additional bits attached to it one is that in between all these standard points there are infinite there's a whole tiny copy of a line in fact for every single standard Point that's the pink bit and each of these infinitesimal neighborhoods which formally is the set of points that's infinitesimally far from a given one and from the middle of these slides you see a definition of infinitesimal is a number which is strictly less than one over k for K one two three and all standard numbers

I'm deliberately not really defining standard and instead here's a quiz that I'm going to give like probably about a minute then I'm going to ask you guys to give answers for real and I'll say don't think too deeply about it and hear Epsilon represents 1 over n and represents some unlimited number and this is 0.999 repeating and this e is the one that's like 2.718 and this is just pi so is pi plus E standard anyone yes okay is Epsilon minus 2 standard okay there's two answers so far I really like this Thumbs Up Down thing okay is tangent of big n standard okay one person gave a thumbs up to that that is not true that's not standard because n is not standard like if it was tangent of like four then it would be a standard expression basically because there's some expression here that involves some basically a funny number one that is not a standard limited real number and I know that that's somewhat circular but the idea really is that you can semantically identify whether something is standard okay the last one is one minus 0.999 repeating standard gotten one so far two three four okay everyone gave this so I think you guys based on the thumbs up do you have the at least the intuitive idea for which I'm very glad let's turn our attention back to this the very top line for a moment notice all the omegas the W looking thingies sorry before you go on could you I guess I sort of intuitively get why the last one is not supposed to be standard but it hinges I guess on what you mean by this expression that just subtract I had wondered if someone would bring that up actually this one I was honestly just going to gauge people's responses if someone asked that I think they have the idea anyway and the quiz fulfills its purpose for this one I guess if I meant anything it's to evoke that no it is really close to but not quite one I know that the notation has like there's ambiguity in that that's interesting though thanks and thank you for bringing up that point okay so the last one is not zero it is some Epsilon because you've got a 0.99 repeating like infinitely yeah or like big n would be repeatedly Big N repeating is more accurate yes but but if I just have the the um normal whatever that is the normal 0.99 this should be zero right yes okay good so it's actually mine is Epsilon is that right um this will be Epsilon yeah sorry excuse me yeah yeah cool and just as these numbers the infinitesimal numbers are smaller than any standard number the unlimited ones are well bigger

the sense in which they're like I had said in the title of the talk infinite numbers that act like they're finite basically the way in which they act like they're finite is that they really do just behave like ordinary numbers for example take that Omega and Omega over two okay here's a question what's Omega minus Omega over 3 or Omega some unlimited number two-thirds Omega yes exactly whereas the usual notion if you just say like Infinity minus infinity unless you know the expression that produces those Infinities that could be anything at all and you usually would just call that undefined but here not only is it defined you have a pretty precise answer you're just doing arithmetic do I have to be scared Omega squared or is that just Omega squid Omega squared is just Omega squared but the square root of Omega squared is omega so they really are behaving like standard numbers in all arithmetic ways and in this case you can even do a sort of induction where you can count down from Omega all the way down to zero or into the sort of standard initial segment is there a difference between this on uh working with finite numbers and then taking the limit after you've performed your computation it is there's a way to compile all of this into statements about the standard real numbers this was actually a bit later but isn't it here's why like the one-liner for why should you care about any of this because any first order statement about the real numbers is true if and only if it's true about these hyper real numbers which is a really broad class of statements like all the ones that are simple arithmetic are ones for example this implies that there are unlimited prime numbers not just that there are infinitely many primes which is kind of trivial but that there's individual unlimited numbers which are prime they have no finite factors they have no infinite factors except for themselves and well I guess they have one as a finite Factor too but that's it and they're also also infinitely many of them because there's infinitely many crimes thanks I will um keep the faith and keep listening cool extremely important point that will be hammered home literally the next slide anyway so just won't do it this line a world in every grain of sand it's from a William Blake poem and that's how I like to think of things I think he's standing in front of a different slide to the one you think oh interesting okay yep now do you see this bit is supposed to capture that as you'd seen here in this neighborhood

that's only really it's the extent of a point it's only one point wide one standard point wide but it contains a whole copy of the line and you can just keep doing this like say I picked this point right here I'm going to do it in green at that point I can zoom in by another Factor let's save Omega or something and then I'm basically looking at an Epsilon squared line and then I can zoom in more to get an Epsilon cubed even tinier line Point by point and this never ends and you can go the opposite direction too with Omega Omega squared cubed Etc Omega to the Omega but then the squared has has different properties than normal numbers the square root like you're squaring you're squaring epsilons to get these totally new lines that's not a normal number thing they only share finitary properties with normal numbers first order ones properties that quantify only over elements of an individual set and not over subsets of some given set that would be a second order property and this property of like having a whole line attached having some set attached to it I think is a second order property I will think more on that one though so thanks for bringing up but I think that's a second order property and therefore it would not be captured by this transfer principle keep in mind that first order logic and first order statements are quite Broad and capture a lot of useful stuff like they can't capture a statement like the the real numbers are complete but these include the real numbers and so they have a form of completeness that's good enough in practice because it's at least as fine as the regular one anyway these fact that these infinitesimal neighborhoods are disjoint gives two really nice things one is if you like if you prefer doing like to prove that a equals B by doing a bunch of algebraic manipulations and getting a whole chain of equalities to finally show that something does indeed equal B versus an analysis you usually prove that a equals B by showing that it's less than it it's greater than it and so you squeeze its value which I've never liked doing I I don't like proving inequalities twice I'd rather just do it with an equality here this relationship of is infinitely close to does the heavy lifting it's basically the equal sign of this sort of infinitely approx infinitely close approximation world does that make no sense okay some okay someone's agreeing it makes no sense that's two with someone like to venture a specific point of confusion my thumbs up was agreeing with you not

that it doesn't make sense oh okay cool another another this one will actually require saying stuff out loud give because they're these neighborhoods are disjoint as well you have the very important fact that every non-standard limited number so say one minus Epsilon is infinitely close to at most one standard real number and therefore can be uniquely rounded off to it basically by dropping the epsilons and that's called the standard part function and you're going to compute it right now what is the standard part of two two yes good what is the standard part of that third expression or Epsilon the 77 minus 3 Epsilon plus 2.7 Epsilon cubed plus 1 minus Epsilon all times Epsilon to the power of n how many levels of inception do you want us to go up 78 . messing with you guys to be honest yes it's 77. the person who just like added the one I see what you did I did that too the first time but yeah like the rabbit hole goes as deep as you're willing to make it go okay last one what is the standard part of n where n is unlimited zero I mean well there's there's no standard part yes it's not zero this one is yeah this is in the category of undefined like if like usually in practice you want your computation to end up with something limited it's fine if it has a bunch of epsilons attached but typically if you're carrying around some infinite term at the end of your computation you've been doing something odd which we're about to do right now yeah we're going to differentiate a function twice in fact which is not even continuous to begin with specifically the step function this particular instance of the step move toward a little bit again okay I'm hiding my own slides it's all right we don't buy it okay so the step function is zero for negative inputs one for positive inputs and one half at the point zero and that curve in red I drew is uh what I call a hyperfinal pro that should be approximation or a hyper finite approximation but the point is that we model it with some function which is differentiable which is infinitely close to it at all the standard points and the one I picked is this any logistic curve would work there are infinitely many possible ways to do this in general usually you just pick one that's convenient and has a nice functional form so I went with something easy just the basic logistic function where n is unlimited that multiplication by n that's highlighted in red here is basically where all the magic happens so the table is a case analysis if x the

first thing means that X is strictly positive and is not infinitesimal also called appreciable can I ask here when you write e to the minus n x what you mean is some kind of limit I guess unless you have a different definition of the exponential so it's or is it just the definition you gave earlier the one I gave earlier and do I have to worry about that some converging or something or like the standard part converges and I just don't worry about the other part of unlimited terms itself always exists there's some saturation principle that guarantees this I see like yeah okay so for positive inputs what will happen is okay plug in some standard positive number for X let's just say 2 because why not then we would take e to the power of negative 2 n is unlimited so this is basically like taking e to the negative Infinity which in this case is not exactly zero but is arbitrarily close to it so we'd end up with one divided by something that's really close to one which itself is infinitely close to one like this function is still strictly monotonically increasing but all of the increases happening in the infinitesible neighborhood is of one and again the fact that infinitesimal neighborhoods are disjoint all these points are will round to one uniquely into no other number and in this way this Curve will just sort of snap onto the straight line similar will hold for the negative case where then you'd end up with the next the same as this except the sign flips so you're dividing by one over something that's Unlimited basically one over n and therefore infinitesimal therefore infinitely close to zero the case of x equal to exactly zero worth using this case to highlight a fact zero is the only number which is both a standard number and infinitesimal and it's the smallest infinitesimal of all like if Epsilon is an infinitesimal of order one and Epsilon squared is ordered to Infinity's order I don't know Omega just multiplying sorry zero is order Omega like multiplying by zero will always just give you zero so this would just be e to the zero which is just one which catches that one standard point that we need in the Very Metal and then what about all this stuff this is the case in the middle of X is infinitely close to zero so it's infinitesimal but it's not exactly equal to zero and the upshot is that it can just be and it will just be this any value between zero and one here I evaluated it at the infinitesimal point one over n so this simplifies to one over one plus

one over e which is about that 0.73 if it was some other infinitesimal I'd say 2 over N I would get some other value but basically this whole cloud of standard and or limited values as soon as I take a standard part the only one that will stay is the standard point at zero which is a half the rest will disappear so I'm doing a bunch of funny stuff but all the funny stuff is done and basically in the gaps between points questions confusions so minus one over n is also um infinitesimally close to zero yes just from Below yeah yeah being infinitely close is more of a close by magnitude foreign is every function uh uh hyper like infinitely close to a continuous differentiable function defined on the hyper reals to continue like basically yeah it's like is it like in the hyper real world true that basically everything is just continuously differentiable I'll have to get back to you on that it does definitely extend the class of differentiability but I haven't looked into it deeply enough to have like strong words to say oh yeah all right how could that be true if the normal real numbers I mean they should behave likely if they used to inside this bigger class right side and I mean you have in mind this kind of these universes like in Topo where somehow for some reason everything is continuous but that's to do with the restricting the class of functions in some way right um mm-hmm no no no it's much more mundane than that you I just mean if you have any non-differential function or not a continuous function on the reals you can extend that to the hyperreels and then find a find a what what is it uh you can find a function that is infinitely close to your original function which is continuously differentiable I think you just get that from Stone virus Strauss or the sort of transfer of that because like the statement that the rationals are dense and the reals you could say that it's star extension so the hyper rationals lovely name that is every real number is infinitely close to some hyper rational number like not necessarily to some rational number but a hyper rational one because standard points can't be infinitely close this is why the idea of infinitely close and being able to cut things into a grid with an unlimited like a specific unlimited number of pieces like and is useful because it lets you implement this idea of the number next to another number even though there is no such thing really certainly not in the standard Wheels because the two numbers are different than whatever distance

they're separated by you can just keep dividing it into but here this hyper finite mesh or sorry this unlimited mesh will just catch everything anyway I had promised differentiating the function not just modeling it and here are its derivatives the first is just the function itself the logistic one the definition of the derivative is basically the same except now Epsilon is a literally infinitely small number and plugging in our function we get so we got these two functional forms which are kind of a mess a better way to think of them I think so on the bottom right picture the one on the left that's supposed to be the direct Delta the idea being it looks like a normal distribution whose variance is infinitesimal or standard deviation is infinitesimal so outside in infinitesimal neighborhood of zero it collapses to being infinitesimal and therefore round to zero and in the infinitesimal neighborhood of zero it has an unlimited height this specific function if you plug in zero I think you get that its height at zero is exactly n over four so it's some infinite number divided by four at zero which is satisfying in a weird way differentiating that again you get the dipole moment sort of how you'd model an idealized tiny magnet and you can just keep differentiating them I did in Wolfram Alpha and they look kind of like flowers and you have the function so I encourage you to try it out it's pretty fun just write these and then just say where n equals like 10 to the 6 or some big-ish number but and then you'll get results that look like this what questions where did the Epsilon go or is it one over n it's the specific Epsilon that's why I put it the very first slide that I meant literally the same one like in the usual setup of a function that's supposed to be standard then you would add a standard part because you're trying to assert that it doesn't matter which Epsilon is picked here I guess it doesn't matter actually for functions like these which are not functions in the usual standard world because there isn't really the vocabulary to coherently talk about them at least not without introducing a bunch of technical machinery um I have a question so um physicists care a lot about direct the direct distribution and also derivatives of that and uh and it works sort of it seems kind of similar to this but the the issue they complain about is that you can't multiply these distributions there is no in general way to just multiply these that's just canonical people I think just pick out convenient

function classes there is a paper on using hyper reels for exactly this I think it's like I can send you a link to it later they use the Frank Palumbo Algebra I think to refer to what formalism they're trying to emulate with the hyper reals but say it's easier are you saying that if I take these derivative these hyper real functions um the derivative the Dirac and these things If I multiply them it doesn't preserve the representative it's like doesn't work well multiplication is that the case it's it depends on the function some of them will work fine others like you can't extend it in a nice way for every function so you end up making concessions through the classes I think that's coming at this from distribution Theory and functional analysis is actually growth index first bit of research around nuclear spaces and stuff which then led to his later work but that's getting kind of far afield so I'd like to get back to this one here are some examples of infinitesimals in that you probably have seen just like behind the hood maybe the top right one is a picture taken from Euclid's elements actually and I absolutely right yeah the top right that's the end of your character a little bit uh sorry say that again I have to move to your left a little bit to the next slide for the orb cam yeah sorry the other way yeah it's a bit too far there is good yeah ah if you can see the change stuff I'm doing right now excuse me I'm stuck in flip blocks fine there I hope I'm not blocking it okay do you see clearly yep that's fun okay so that angle that's formed by F A and E that's called a horn angle and the idea being it's an angle smaller than any angle but it's still there and it's not quite zero I think it's called a horn angle because horns curve like that and come to a point here now we can just say that okay there's an angle and it's infinitesimal it's Epsilon why not something small then the second thing you have a closed unit interval and you just throw a dart what's the probability of hitting the number exactly hitting the number 0.3 foreign nice yeah you really anticipated it yes Epsilon the idea being that yeah you would so each of these points have a tiny line around them okay well that's good we can cut this whole thing up into infinitesimal intervals and then but wait why is it why is it this Epsilon and not one over M for some other M like in it could be that's why doing things like this is like all of them will round to the probability being zero in the standard

sense because all the probabilities that are assigned to any interval containing a standard point no matter what you make it will still be infinitesimal so yeah it well could be two Epsilon or else can I make a critique then maybe this is unfair but so say generalized functions or distributions is a way of layering some more general theory on top of normal functions in a way that accommodates derivatives of direct deltas and so on and it's maybe artificial but at the end of the day we care about the numbers we get out by adopting this artificial ambient theory of the infinite and you're arguing that hyperreals are somehow better because they do away with that somewhat artificial surrounding ambient stuff and a sort of more direct but there's still this Ambience surroundings from the void yeah and it's just like are you just kind of just replacing one set of baggage with a different set of baggage and in that case why is it somehow better I would argue that the sort of meta reason that I like these so much and think that they're better is that I see them as the next step natural step in the enlargement of the idea of numbers like we all start with not even the full set of Naturals just one two three to begin with and then at some point you realize you can just keep adding one then you get to confront the idea of infinity at some point zero is introduced then negatives usually fractions have already been introduced at this point but then if you want to take a square root you need irrationals and then if you want to solve algebraic equations you need complex numbers here if you want to distinguish scenarios like these you need infinitely small and large numbers and each one is a clean enlargement of the other like the typical upshot of how you actually use these in practice and why the arbitrariness of them in most scenarios is not actually a problem is that for scenarios that do have some standard interpretation then there will be an independence from which particular infinite or infinitesimal number is picked in non-standard scenarios there is more choosing but I to my understanding that happens in the ambient standard Theory as well this one allows us to draw much more strong intuitions about numbers which are maybe like the strongest mathematical intuitions of all except for like basic geometry and shapes it also it also led to swap limits right like that's something we couldn't do before with derivatives yeah like it gives more sequences a limit including plenty that weren't convergent

before that's like here you can compile all of these hyper real numbers into Infinite sequences of real numbers and that's not especially constructive but the reels themselves are not really computable either so I don't worry too much about that I feel like the the transfer principle is doing a lot of heavy lifting for you conceptually would that be accurate I mean when I think about this and I I guess I am to some degree convinced this is a useful way of thinking about it uh the the role of the transfer principle and first order logic here seems to be a while both a blessing and a curse it's a blessing in that some sense you can't say at least if you restrict the first order logic these are as good as the real numbers you think you like I think of it like this like okay say this green box is just this whole standard mathematical Universe like sets functions points all that bunch of arrows for all of them okay now you create this hyper thingy which gives you all this extra room but there's also this intermediate region where transfer carries stuff the kind of questions that live out in the boundaries of this fuzzibly drawn analogy or like if I just give you two unlimited numbers just out of the Void Like A and B with no further information a question like is a odd or even is it greater than b basically any question at all except that it's bigger than any standard number can't be answered it doesn't carry any information other than the fact that it's big and it's kind of a placeholder for big however that's only in the purely external Realm things that stay within this first order realm will transfer in a unique way and this outer realm is basically where research in the overall field happens I think the people who tend to use this in practice are people who work in differential equations Terence Tower uses it a lot and ultrafinitis Ultra finite tests a weirdly large amount of them like this I think because it sort of discretizes the continuum in fact as an example of this discretization I'd like you to turn your attention to the two circles on the bottom right oh polygon and circle looking thing this is to illustrate a concept that's called hyper-finite approximation where okay as a basic principle given some smooth curve you can imagine it being made okay there's a bunch of points but now replace the point with AN infinitesimal line around it and now this curve has become some infinitesimal chain with an unlimited number of links and therefore well it's not a polygon because it

hasn't closed up but it is made of lines and this circle if I just cut it every point on it is an infinitely small line in total it will have some unlimited number of lines like here's one sense in which those numbers are not arbitrary we touched on this earlier but I'll make this more explicit say that the number of points to identify the number of sides of this circle is some unlimited number c then a circle that had twice the radius which is also an infinite Circle keep in mind because the radius of this would be well it's like 2 pi R equals c so this would be C over 2 pi so this radius is also Unlimited but anyway circle with twice the radius would have two c many side pieces possibly plus or minus one there might be some roundoff and so like one of these numbers you you do end up making up a name out of the Void but typically it's links to other quantities that are unlimited are not arbitrary and you can write those down and given those relations you do have a proper sizing for the Infinities except for the first one that was introduced that sort of makes sense you have a question about this example but uh maybe I can save it until the end or do you have more to say or you want to finish there no I decided to leave it enough time for questions so I'm pretty satisfied this gives like 15 minutes so fire away after it yeah I'm quite interested in this this final example you gave because uh well given you mentioned Euclid so the left hand example I guess in modern day I would think of it as being just that set of points or perhaps the set of lines but Euclid would have thought of it as a construction and the analogy between constructions and programs and proofs of course goes back to Euclid so you could think of that that polygon is actually the construction that produces it as a sequence of deductions in Euclid's axioms but then I wonder is there such a thing as a hyper finite proof right I mean the construction of the right hand side is in some sense a limit of the constructions on the left hand side is there like a developed theory of hyper finite proofs like taking some finite quantity and I'm not quite not quantity I mean constructions so Euclid doesn't think of the polygon as I mean he doesn't think of numbers right he thinks of the construction so and the construction is like an algorithm so I'm kind of asking have have people in computer science the closest I've seen is that Nick Bostrom seems to know this stuff because he mentions the use of hyper finite

numbers for infinite ethics if the universe is actually a fair number two Sylvia van mackers is one in particular the is there any computers yeah I get a couple remarks on this yeah um so one thing is if you make models of these hyper reels it uses um it uses Ultra filters so it uses a weak form of vaccine of choice so that makes it all extremely um kind of problematic a little bit from like actual constructing things um there are also syntactic approaches where you don't do that but uh but yeah this this is sort of an issue um there is an existing theory of constructive uh hyper rails I'm not sure this is quite subtle because no no I understand I understand but um uh closest I know of for oh for a hyper finite proof I think these proofs are doing essentially the same thing but they're not using this formalism proving like some property like I've read this in the context of finding math apps so this time not speaking with authority and you really should just like Google this one but they say that they prove that at some finite numbers the problem is undecidable and you the way they prove it is by proving it at infinity and that there's some transfer and the way they're talking about it they're being these increasing sequences of finite numbers and gives the whole thing a similar feel I think if you saw it you'd be able to suss it out I can send you a link yeah thanks I'll ask you about that later I don't know if anyone in computer science computer science has done anything with it close this one knows like Dan Roy and house said to one who who have used it in statistics especially based statistics I guess too put some red meat in front of Alexander it makes you wonder if there's like a hypophonite version of the theory of linear orders that contains a model of cyclic orders that would be maybe version of how you're thinking of like the like the integers of the cyclic group is kind of being linear too but also cyclic in a way or is this unrelated um yeah I'm not sure I think I'll open the floor for other people to ask questions rather than hug you for the next 10 minutes other questions for um yeah also people want more examples I can give um while other people are thinking I'll ask another one so what got you into this uh cult I mean slash uh system of beliefs it feels a bit to me like the the geometric algebra uh what uh community I think the main reason being like yeah he knows well that's the first time we had met was talking about a people that him and James Cliff had done on in

this vein the way it unifies discreet and continuous the theme of enlargening Number systems which is one of my favorite things about math because the concept of a number compresses so much information the really Elementary way you can work with things I love that you can just do the most naive sounding thing of just draw an infinitely big box you can literally just say that and that's totally rigorous and it just works and that so many statements about sequences that are awkward just become these nice idealized things and then unlike most Cults the cult most Cults don't have a transfer principle that's that's an awesome real at least up to finitary statements about them you're sort of compelled to if not accept these then not like totally dismiss them they have because of the if and only if which is also not obvious yeah I would agree with that you also give a sort of different view of infinity that's closer in line with people's intuition a little part whole relation like for example if you had a an open versus a closed square with or without the boundary you could say that by some infinitesimal amount the closed squares strictly larger than the open one in the theme of not throwing away information basically you also told me about radically elementary probability Theory oh right yeah there's this probability Theory Book that uses this stuff not even very much of it it really only uses the concepts of infinitely large and small and that's it and it covers in just 80 Pages like three times more probability Theory than I've done in my whole life yeah I have to admit I'm a little skeptical of that it probably is just more or less the same as skating over the details no uh rather than making using a radically simpler rigorous formalism is it is that a rigorous treatment I'm okay with a not so rigorous treatment of Elementary probability but false advertising he sets up the whole the whole whole formalism in appendix a he says that he wrote the appendix to be self-destructing everything in it is rigorous his Arguments for them is sometimes a little rushed that's for sure like it's not easy reading but even if it was fully expanded out I think it would still be under 200 Pages like safely and does cover quite a lot of material like it gets deep into stochastic processes and there's another book on stochastic calculus that uses it as well and I've taken a look at that one the real benefits are that you don't need measure Theory because now the only event that has probability zero is the

empty event which is impossible to condition on because it can't happen and that alone eliminates so many problems like just it's fine if you divide by your five Epsilon also measuring things becomes just counting again like you can define a generalized measure it's just a weighted counting scheme and it also lets you treat discrete and continuous at the same time and that book does so as well we're a continuous distribution is of quote finite distribution by using this idea of a hyper finite set Counting combinatorics what does that mean yo what does counting combinators mean and then just look up I'm not sure but I ever another question so yeah so in in linear logic there's a also a very important role for the the different kinds of infinite so an infinite that is sort of potential and only realize through interaction in the sense that an unlimited supply of something is like a box you can keep drawing from and it never runs out but it's not infinitely large in some sense um got it so is do you see any I mean okay so that's a important idea of linear logic is there something like this yeah is there another Infinity like they're an actual Infinity linear logic as well not just the potential authentically yes but that's part of the tricky part of trying to understand what the content of linealogic is mathematically so I think that's a hard question to answer okay but it's not something like this idea of something well not being limited that you have as much as you want but it's ultimately some finite amount it's just bigger than any other number you had named before and whatever you're talking about and if it's some first order thing you can quantify over the elements you're talking about and just pick something bigger than all of them and all possible combinations of them up to some finite depth of symbols or something basically through compactness theorem yeah the reason I ask is this there's some very similar kind of flavor to some part of that but I don't know how to say anything more precise so yeah maybe it's a vague question but I find it very interesting I mean this trying to put some mathematical detail on this potential infinite is what makes linear logic kind of interesting so I mean one thing that sort of intuitively that they try to capture is a vague property like one example I thought about on a car ride is how there are many forms of cancer like the ultimate number of forms that can happen is sort of limited by the discreetness of the universe but is